As a conclusion, I would like to recapitulate all the bifurcations I have meet
in this project.
The figure 3.6 allows us to have a more syntetic vue of the bifurcations
of the KSE.

Figure 3.6:
Bifurcation diagram

In fact, I have seen that, even if for low value of , the theoretical study
can predict the dynamic of the Kuramoto-Sivashinsky, the more increase, the
more complex the dynamic is.
The dynamic of the KSE is a big problem because this equation is representative
of lot of phenomenon like :

chemical oscillation or chemical fronts

flame front in combustion

numerical dissipation

That's why this problem is very studied.

Moreover, for the course of hydrodynamic instabilities, this equation was very
interesting. This project allows me to see the applications of this course and
make me work on a important part of my formation : the numerical methods.